The phrase "a second or even" is correct and usable in written English.
It can be used to indicate an additional or alternative option that is more extreme or significant than the first mentioned.
Example: "You might need a second or even a third opinion before making a decision."
Alternatives: "a second or possibly" or "a second or perhaps".
The problem has sometimes been that there has not been a second or even third choice identified.
It would not be until a second or even a third bad test that they would seriously be considered to be forced out of the college.
The patient must then return to regular maintenance treatment with an artificial kidney but may receive a second or even a third graft.
I know many students who failed the first time and had to take it a second or even a third time.
In the US, minimum wage employees do not earn enough to survive and are forced to take a second or even a third job to make ends meet.
He was the ruling party's compromise candidate, a second or even third choice after the main contenders cancelled each other out.
Trump spreads lies the way terrorists plant bombs: one goes off, and when the first responders rush in, there's a second, or even a third.
Luise, Germany When one or both of a couple marry for a second or even third time, friends who celebrated and gave gifts to honor the first occasion shouldn't be obligated to send another wedding gift.
Whatever is happening to the patients, it seems to be temporary: many CCSVI patients return for a second (or even a third) operation several months after their initial procedures.
Attention armchair sleuths: After viewing the Italian psychological thriller "The Double Hour," you may want to see it a second or even a third time to decipher its secrets.
In desperation, he sometimes loads the link into a second or even a third browser tab as well, and bombards the website's server with multiple requests for the page.
